Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs

We study isoperimetric inequalities on "slabs", namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension-one base. As our two main applications, we consider the case when the base is the flat torus $\mathbb{R}^2 / 2 \mathbb{Z}^2$ and the standard Gaussian measure in $\mathbb{R}^{n-1}$. The isoperimetric conjecture on the three-dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to $0$, $1/2$ and $1$ by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths $(\beta,1,1)$ in a new range of relatives volumes $\bar v \in [0,1/2]$. In particular, we confirm the conjecture for the standard cube ($\beta=1$) for all $\bar v \leq 0.120582$, when $\beta \leq 0.919431$ for the entire range where spheres are conjectured to be minimizing, and also for all $\bar v \in [0,1/2] \setminus (\frac{1}{\pi} - \frac{\beta}{4},\frac{1}{\pi} + \frac{\beta}{4})$. When $\beta \leq 0.919431$ we reduce the validity of the full conjecture to establishing that the half-plane $\{ x \in [0,\beta] \times [0,1]^2 \; ; \; x_3 \leq \frac{1}{\pi} \}$ is an isoperimetric minimizer. We also show that the analogous conjecture on a high-dimensional cube $[0,1]^n$ is false for $n \geq 10$. In the case of a slab with a Gaussian base of width $T>0$, we identify a phase transition when $T = \sqrt{2 \pi}$ and when $T = \pi$. In particular, while products of half-planes with $[0,T]$ are always minimizing when $T \leq \sqrt{2 \pi}$, when $T > \pi$ they are never minimizing, being beaten by Gaussian unduloids. In the range $T \in (\sqrt{2 \pi},\pi]$, a potential trichotomy occurs.